Ajay Dulichand Borkar¹, Dipjyoti Nath¹, Sachin Singh Gautam^1,*

The International Conference on Computational & Experimental Engineering and Sciences, Vol.32, No.2, pp. 1-1, 2024, DOI:10.32604/icces.2024.011404

Abstract In recent years, machine learning (ML) has emerged as a powerful tool for addressing complex problems in the realms of science and engineering. However, the effectiveness of many state-of-the-art ML techniques is hindered by the limited availability of adequate data, leading to issues of robustness and convergence. Consequently, inferences drawn from such models are often based on partial information. In a seminal contribution, Raissi et al. [1] introduced the concept of physics informed neural networks (PINNs), presenting a novel paradigm in the domain of function approximation by artificial neural networks (ANNs). This advancement marks a… More >

Z. D. Han¹, S. N. Atluri^2,3

CMES-Computer Modeling in Engineering & Sciences, Vol.97, No.3, pp. 199-237, 2014, DOI:10.3970/cmes.2014.097.199

Abstract This paper presents a new method for the computational mechanics of large strain deformations of solids, as a fundamental departure from the currently popular finite element methods (FEM). The currently widely popular primal FEM: (1) uses element-based interpolations for displacements as the trial functions, and element-based interpolations of displacement-like quantities as the test functions; (2) uses the same type and class of trial & test functions, leading to a Galerkin approach; (3) uses the trial and test functions which are most often continuous at the inter-element boundaries; (4) leads to sparsely populated symmetric tangent stiffness… More >

Z. D. Han¹, S. N. Atluri^2,3

CMES-Computer Modeling in Engineering & Sciences, Vol.97, No.1, pp. 1-34, 2014, DOI:10.3970/cmes.2014.097.001

Abstract The concept of a stress tensor [for instance, the Cauchy stress σ, Cauchy (1789-1857); the first Piola-Kirchhoff stress P, Piola (1794-1850), and Kirchhoff (1824-1889); and the second Piola-Kirchhoff stress, S] plays a central role in Newtonian continuum mechanics, through a physical approach based on the conservation laws for linear and angular momenta. The pioneering work of Noether (1882-1935), and the extraordinarily seminal work of Eshelby (1916- 1981), lead to the concept of an “energy-momentum tensor” [Eshelby (1951)]. An alternate form of the “energy-momentum tensor” was also given by Eshelby (1975) by taking the two-point deformation gradient tensor… More >

J. D. Hales¹, S. R. Novascone¹, R. L. Williamson¹, D. R. Gaston¹, M. R. Tonks¹

CMES-Computer Modeling in Engineering & Sciences, Vol.84, No.2, pp. 123-154, 2012, DOI:10.3970/cmes.2012.084.123

Abstract The equations governing solid mechanics are often solved via Newton's method. This approach can be problematic if the Jacobian determination, storage, or solution cost is high. These challenges are magnified for multiphysics applications. The Jacobian-free Newton-Krylov (JFNK) method avoids many of these difficulties through a finite difference approximation. A parallel, nonlinear solid mechanics and multiphysics application named BISON has been created that leverages JFNK. We overview JFNK, outline the capabilities of BISON, and demonstrate the effectiveness of JFNK for solid mechanics and multiphysics applications using a series of demonstration problems. We show that JFNK has More >

Sunilkumar N¹, D Roy^1,2

CMES-Computer Modeling in Engineering & Sciences, Vol.66, No.3, pp. 249-284, 2010, DOI:10.3970/cmes.2010.066.249

Abstract We propose a family of 3D versions of a smooth finite element method (Sunilkumar and Roy 2010), wherein the globally smooth shape functions are derivable through the condition of polynomial reproduction with the tetrahedral B-splines (DMS-splines) or tensor-product forms of triangular B-splines and 1D NURBS bases acting as the kernel functions. While the domain decomposition is accomplished through tetrahedral or triangular prism elements, an additional requirement here is an appropriate generation of knotclouds around the element vertices or corners. The possibility of sensitive dependence of numerical solutions to the placements of knotclouds is largely arrested… More >

Amit Shaw¹, B. Banerjee¹, D Roy^1,2

CMES-Computer Modeling in Engineering & Sciences, Vol.26, No.1, pp. 31-60, 2008, DOI:10.3970/cmes.2008.026.031

Abstract A generalization of a NURBS based parametric mesh-free method (NPMM), recently proposed by Shaw and Roy (2008), is considered. A key feature of this parametric formulation is a geometric map that provides a local bijection between the physical domain and a rectangular parametric domain. This enables constructions of shape functions and their derivatives over the parametric domain whilst satisfying polynomial reproduction and interpolation properties over the (non-rectangular) physical domain. Hence the NPMM enables higher-dimensional B-spline based functional approximations over non-rectangular domains even as the NURBS basis functions are constructed via the usual tensor products of… More >

S. N. Atluri¹, H. T. Liu², Z. D. Han²

CMES-Computer Modeling in Engineering & Sciences, Vol.15, No.1, pp. 1-16, 2006, DOI:10.3970/cmes.2006.015.001

Abstract The Finite Difference Method (FDM), within the framework of the Meshless Local Petrov-Galerkin (MLPG) approach, is proposed in this paper for solving solid mechanics problems. A "mixed'' interpolation scheme is adopted in the present implementation: the displacements, displacement gradients, and stresses are interpolated independently using identical MLS shape functions. The system of algebraic equations for the problem is obtained by enforcing the momentum balance laws at the nodal points. The divergence of the stress tensor is established through the generalized finite difference method, using the scattered nodal values and a truncated Taylor expansion. The traction More >

A.I.Tolstykh, D.A. Shirobokov¹

CMES-Computer Modeling in Engineering & Sciences, Vol.7, No.2, pp. 207-222, 2005, DOI:10.3970/cmes.2005.007.207

Abstract A way of using RBF as the basis for PDE's solvers is presented, its essence being constructing approximate formulas for derivatives discretizations based on RBF interpolants with local supports similar to stencils in finite difference methods. Numerical results for different types of elasticity equations showing reasonable accuracy and good$h$-convergence properties of the technique are presented. Applications of the technique to problems with non-self-adjoint operators (like those for the Navier-Stokes equations) are also considered. More >